3.853 \(\int \frac{\sqrt{1+x}}{\sqrt{1-x} \sqrt{x}} \, dx\)

Optimal. Leaf size=10 \[ 2 E\left (\left .\sin ^{-1}\left (\sqrt{x}\right )\right |-1\right ) \]

[Out]

2*EllipticE[ArcSin[Sqrt[x]], -1]

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Rubi [A]  time = 0.0259154, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ 2 E\left (\left .\sin ^{-1}\left (\sqrt{x}\right )\right |-1\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[1 + x]/(Sqrt[1 - x]*Sqrt[x]),x]

[Out]

2*EllipticE[ArcSin[Sqrt[x]], -1]

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Rubi in Sympy [A]  time = 2.99296, size = 10, normalized size = 1. \[ 2 E\left (\operatorname{asin}{\left (\sqrt{x} \right )}\middle | -1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1+x)**(1/2)/(1-x)**(1/2)/x**(1/2),x)

[Out]

2*elliptic_e(asin(sqrt(x)), -1)

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Mathematica [C]  time = 0.26948, size = 104, normalized size = 10.4 \[ \frac{2 \sqrt{\frac{x-1}{x+1}} \sqrt{\frac{x+1}{x-1}} \left (\sqrt{x-1} \sqrt{\frac{x+1}{x-1}} x+\frac{i \sqrt{2} x E\left (i \sinh ^{-1}\left (\frac{\sqrt{2}}{\sqrt{x-1}}\right )|\frac{1}{2}\right )}{\sqrt{\frac{x}{x-1}}}\right )}{\sqrt{-(x-1) x}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[Sqrt[1 + x]/(Sqrt[1 - x]*Sqrt[x]),x]

[Out]

(2*Sqrt[(-1 + x)/(1 + x)]*Sqrt[(1 + x)/(-1 + x)]*(Sqrt[-1 + x]*x*Sqrt[(1 + x)/(-
1 + x)] + (I*Sqrt[2]*x*EllipticE[I*ArcSinh[Sqrt[2]/Sqrt[-1 + x]], 1/2])/Sqrt[x/(
-1 + x)]))/Sqrt[-((-1 + x)*x)]

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Maple [B]  time = 0.016, size = 39, normalized size = 3.9 \[ 2\,{\frac{\sqrt{2}\sqrt{-x} \left ({\it EllipticF} \left ( \sqrt{1+x},1/2\,\sqrt{2} \right ) -{\it EllipticE} \left ( \sqrt{1+x},1/2\,\sqrt{2} \right ) \right ) }{\sqrt{x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+x)^(1/2)/(1-x)^(1/2)/x^(1/2),x)

[Out]

2*2^(1/2)*(-x)^(1/2)*(EllipticF((1+x)^(1/2),1/2*2^(1/2))-EllipticE((1+x)^(1/2),1
/2*2^(1/2)))/x^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x + 1}}{\sqrt{x} \sqrt{-x + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x + 1)/(sqrt(x)*sqrt(-x + 1)),x, algorithm="maxima")

[Out]

integrate(sqrt(x + 1)/(sqrt(x)*sqrt(-x + 1)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{x + 1}}{\sqrt{x} \sqrt{-x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x + 1)/(sqrt(x)*sqrt(-x + 1)),x, algorithm="fricas")

[Out]

integral(sqrt(x + 1)/(sqrt(x)*sqrt(-x + 1)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x + 1}}{\sqrt{x} \sqrt{- x + 1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1+x)**(1/2)/(1-x)**(1/2)/x**(1/2),x)

[Out]

Integral(sqrt(x + 1)/(sqrt(x)*sqrt(-x + 1)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x + 1}}{\sqrt{x} \sqrt{-x + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x + 1)/(sqrt(x)*sqrt(-x + 1)),x, algorithm="giac")

[Out]

integrate(sqrt(x + 1)/(sqrt(x)*sqrt(-x + 1)), x)